Find a primitive generator for $\displaystyle \mathbb{Q}(\sqrt2,\sqrt3,\sqrt5)$ over $\displaystyle \mathbb{Q}$.

I found the polynomial for the root $\displaystyle \alpha=\sqrt2+\sqrt3

+\sqrt5$, which is $\displaystyle x^8-40x^6+352x^4-960x^2+576$. But I'm having trouble showing the polynomial is irreducible. Some help please.